等差数列、等比数列的一个新的性质

结论1已知等差数列{an},r,s,t是互不相等的正整数,则有(r-s)at (s-t)ar (t-r)as=0.证明设等差数列{an}的首项为a1,公差为d,则(r-s)at (s-t)ar (t-r)as=(r-s)a1 (t-1)d] (s-t)a1 (r-1)d] (t-r)a1 (s-1)d]=(r-s) (s-t) (t-r)]a1 (r-s)(t-1) (s-t)(r-1) (t-r)(s-1)]d=(r-s) (s-t) (t-r)]a1 (rt-st-r s) (sr-tr-s t) (ts-rs-t r)]d=0.此结论可以在知道等差数列中的任意两项的情况下,求出第三项的值.比如问题:已知数列{an}为等差数列,ap=q,aq=p,且p≠q,求ap q.略解由结论1可知,(p-q)ap q q-(p q)]ap (p q)-p]aq=0,即(p-q)ap q-pq pq=0,…